Every element of a finite field is expressible as a sum of two squares
This result is attributed to Henry Mann.
Suppose is a finite field. Then, every element can be expressed in the form , where .
- Multiplicative group of a finite field is cyclic: We actually only need the weaker statement that, for a field of odd characteristic, exactly half the elements of the multiplicative group are squares.
- Product of subsets whose total size exceeds size of group equals whole group: If are subsets of a finite group , where , then .
Case of characteristic two
In this case, the square map is surjective and every element is a square, because the multiplicative group is of odd order.
Case of odd characteristic
- Reasoning in the multiplicative group: Suppose has elements. Then its multiplicative group has elements. By fact (1), the multiplicative group is cyclic of order , which is even. Thus, exactly half the elements (corresponding to even powers of the generator) are squares. Since is also a square, we obtain elements of that are squares.
- Reasoning in the additive group: We now apply fact (2) with as the additive group of , which has size , and both and as equal to the set of (multiplicative) squares, which has size .